Stresses Due to a Sharp Notch in a Work-Hardening Elastic-Plastic Material Loaded by Longitudinal Shear

Abstract

A work-hardening elastic-plastic stress analysis is presented for a sharp notch or, as a limiting case, a crack perturbing a remotely applied uniform stress field. Mathematical complexities are reduced through considering the kinematically simple case of antiplane longitudinal shear deformations and by employing a deformation plasticity theory rather than the more appropriate incremental theory. Consequently, a general solution is available valid for any relation between stress and strain in the work-hardening range, so long as the remotely applied stress does not exceed the initial yield stress. When a power law relates stress to a strain in the work-hardening range, the deformation theory solution is also the correct incremental solution at low applied stress levels causing yielding on a scale small compared to notch depth. For cracks, the near crack tip strain field is shown to depend on loads and geometry only through the elastic stress intensity factor when yielding is on a small scale, and the elastic-plastic boundary and lines of constant strain magnitude are circles. Extensive numerical results are tabulated for a crack, 45 deg V-notch, and 90 deg V-notch in power-law-hardening materials, and exhibited graphically for a crack.